R.B Warfield

The influence of ideal structure on representation theory

The purpose of this paper is to explore the representation theoretic consequences of the two-sided ideal structure of a Noetherian ring, with special emphasis on an additional combinatorial structure on the prime spectrum of the ring, the graph of links. We obtain existence theorems for extensions, and, for rings satisfying the “strong second layer condition,” we obtain extensive information about the indecomposable injective modules. Particularly detailed results are obtained when the ring is polynormal.

The number of generators of a module over a fully bounded ring

We prove a theorem giving an upper bound for the number of generators of a module in terms of local data, for a class of rings which includes Noetherian rings which are right fully bounded. Forster and Swan have proved such results for commutative Noetherian rings and finite algebras over these rings. Eisenbud and Evans have shown, further, that from any given set of generators for a finitely generated module over such an algebra, the number predicted by the theorem can be obtained by a certain kind of elementary transformation. This result we also obtain for our wider class of rings. Regarding these results as results about the existence and structure of epimorphisms F -+ A, where F is free and A the module, we obtain also corresponding results were F is a direct sum of copies of some projective module other than R. The chief difficulties throughout are those involved in finding substitutes for arguments involving localization, which is not an available tool in our context. For prime rings of Krull dimension one, we can remove the boundedness condition, and also obtain for hereditary Noetherian prime (HNP) rings the analogue of a result of Roiter’s for lattices over orders. We also obtain a weaker result for epimorphisms F + A, where F is a direct sum of copies of some finitely presented module which is not projective, and a uniqueness result (up to right equivalence) for epimorphisms F + A, where F is a large enough free module. In the following, terms like “Noetherian” and “Goldie”-&e applied on both sides unless specified otherwise. If A is a module, we let the direct sum of n copies of A be denoted wi (rather than A”), in or&r to avoid confusion when referring to ideals. If A is a module, we let g(A) be the smallest number which is the cardinality of a set of generators for A. A prime ring is right bounded if every essential right ideal contains a nonzero two-sided ideal, and a ring R is right fully bounded if for every prime ideal P, R/P is right

KRULL AND GLOBAL DIMENSIONS OF FULLY BOUNDED NOETHERIAN RINGS

The main result of this paper states that the Krull dimension of a fully bounded Noetherian ring containing an uncountable central subfield is bounded above by its global dimension, provided that the latter is finite. The proof requires some results on projective dimensions and on localization (Corollary 4 and Theorem 11, respectively), which may be of independent interest. If P is a prime ideal in a Noetherian ring R, then P is contained in a unique clique, X, a subset of Spec(u) defined below (Definition 6). Now in some circumstances, the set C(X) of elements of R regular modulo every element of X is an Ore set in R, and the localized ring Rx obtained by inverting the elements of C(X) has certain desirable properties. In this case, X is said to be classical (Definition 7). We prove in Theorem 8 that if R is a Noetherian ring of finite global dimension whose cliques are classical, then the classical Krull dimension of R is bounded above by its global dimension. Generalizing work of B. J. Mueller and A. V. Jategaonkar (16, 13), we show that if R is a Noetherian fully bounded ring containing an uncountable set F of central units such that the difference of two distinct elements of F is still a unit, then all cliques in Spec(ii) are classical (Theorem 11). This applies, in particular, if R has an uncountable central subfield, as in the abstract. Using Theorems 8 and 11 in their general forms, K. R. Goodearl and L. W. Small have shown that the inequality of Krull and global dimensions is true for all Noetherian P. I. rings (10). In this paper, all modules are right modules unless it is indicated otherwise. If M is a module, then r(M) is the right annihilator of M and (if appropriate) l(M) is the left annihilator of M. A ring is Noetherian if it satisfies the ascending chain condition on right and left ideals. If P is a prime ideal of R, then R/P is right bounded if every essential right ideal of R/P contains a nonzero two-sided ideal. A ring R is fully bounded Noetherian (abbreviated FBN) if it is Noetherian and for every prime ideal P, R/P is both right and left bounded. Part of this research was done while the first author held a visiting position at the University of Washington. He is grateful to that institution for its hospitality. The research of the second author was supported in part by a grant from the NSF. 1. Homological results. LEMMA 1. Let R be a ring, A a right R-module, and M a maximal co-Artinian ideal of R. Then for every nonnegative integer n, ExtR(A,R/M) — 0 if and only

Affiliated series and extensions of modules

The main purpose of this paper is to examine the influence on the representation theory of a noetherian ring of the (prime) ideal structure of the ring. The results concern the affiliated primes of a module (defined below). We show how the affiliated primes are connected with the annihilator primes and we prove a uniqueness theorem for affiliated primes under certain conditions. We then proceed to analyse what affiliated primes can appear in terms of the ideal structure of the ring, with special reference to the fundamental primes of injective modules. Our starting point is a theorem of Jategaonkar which analyses the structure of modules with exactly two affiliated primes, and which already has consequences for uniserial modules of length two. We give a generalization of this result for modules with longer affiliated series and, in particular, for uniserial modules of length greater than two. In particular, we show that some of the possibilities one might have envisioned in the light of Jategaonkar’s theorem cannot in fact occur and we demonstate a somewhat unexpected connection between affiliated primes and annihilator primes. To put all this material in context, we begin section one with an analysis of what it all means for uniserial modules of length three. We then turn to the uniqueness question for affiliated primes. Here and in the rest of the paper we find it desirable to have some kind of dimension 164 0021-8693/91

Quotient rings and localization for Noetherian rings

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p-Adic and p-cotorsion completions of nilpotent groups

Abstract Criteria are obtained for the localizability of a Noetherian ring at a semiprime ideal S, particularly when S is the nil radical of R. The results are applied to the existence of quotient rings. A decomposition theorem is obtained for certain Noetherian rings which admit Artinian quotient rings.

Serial rings and finitely presented modules

The theory of cotorsion Abelian groups is extended to the category of nilpotent groups, and applications are given to the theory of p-adic completions and exactness properties of the p-adic completion functor. The p-cotorsion completion functor was first extended from Abelian to nilpotent groups by Bousfield and Kan, who defined and studied it using topological methods. We develop the theory group-theoretically, and additional results and applications.